Improved Distributed Algorithms for Exact Shortest Paths

Computing shortest paths is one of the central problems in the theory of distributed computing. For the last few years, substantial progress has been made on the approximate single source shortest paths problem, culminating in an algorithm of Becker et al. [DISC' 17] which deterministically computes (1+ o(1))-approximate shortest paths in (O) over tilde (D + v n) time, where D is the hop-diameter of the graph. Up to logarithmic factors, this time complexity is optimal, matching the lower bound of Elkin [STOC' 04]. The question of exact shortest paths however saw no algorithmic progress for decades, until the recent breakthrough of Elkin [STOC' 17], which established a sublinear-time algorithm for exact single source shortest paths on undirected graphs. Shortly after, Huang et al. [FOCS' 17] provided improved algorithms for exact all pairs shortest paths problem on directed graphs. In this paper, we present a new single-source shortest path algorithm with complexity (O) over tilde (n3/4D1/4). For polylogarithmic D, this improves on Elkin's (O) over tilde (n5/6) bound and gets closer to the (O) over tilde (n1/2) lower bound of Elkin [STOC' 04]. For larger values of D, we present an improved variant of our algorithm which achieves complexity (O) over tilde n3/4+ o(1) + min{n3/4D1/6, n6/7} + D, and thus compares favorably with Elkin's bound of (O) over tilde (n5/6 + n2/3D1/3 + D) in essentially the entire range of parameters. This algorithm provides also a qualitative improvement, because it works for the more challenging case of directed graphs (i. e., graphs where the two directions of an edge can have different weights), constituting the first sublineartime algorithm for directed graphs. Our algorithm also extends to the case of exact.-source shortest paths, giving a complexity of (O) over tilde min n.1/2n3/4+ o(1) +.1/3n3/4D1/6,.3/7n6/7 o + D (O) over tilde For moderately small. and D, this improves on the (O) over tilde(.1/2n3/4 + n) bound of Huang et al.